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# Complex Symmetry of Truncated Composition Operators

• Project Type: Directed
• Directed Project Contributors: Project Contributors: Ruth Jansen, Rebecca K. Rousseau

## Purpose / Abstract

An N x N matrix is a representation of a linear function, called an operator, from RN to itself. For a given operator, each different basis representation of RN creates a different N x N matrix that represents that same operator. An operator is called complex symmetric if there is some basis for which the matrix representation is symmetric (possibly with both real- and complex-valued entries). This idea has also studied on infinite vector spaces such as the Hardy space and operators on those spaces, such as composition operators [2, 4]. Here, we define a truncated composition operator by considering only a finite-dimensional portion of a composition operator, so that we can represent it with a matrix and investigate any resulting complex symmetry.

## Introduction / Background

Symmetric matrices, whose entries form a mirror image across the diagonal, are a familiar concept in the study of linear algebra and its applications. The concept of a complex symmetric operator is similar in nature and has been shown to have a wide variety of interesting applications within operator theory (Garcia, Putinar ). An operator is complex symmetric if it has a symmetric matrix representation with respect to some orthonormal basis. This is distinct from a self-adjoint matrix, in which entries across the diagonal are the complex conjugate of one another rather than identical.

## Resources / Links

 Levon Balayan and Stephan Ramon Garcia. (2010). Unitary Equivalence to a Complex Symmetric Matrix: Geometric Criteria. Operators and Matrices, 4(1): 53-76.

 Stephen Ramon Garcia and Mihai Putinar. (2005). Complex Symmetric Operators and Applications. Transactions of the American Mathematical Society 358(3): 1285-1315.

 Carl Cowen and Barbara MacCluer. (1995). Composition Operators on Spaces of Analytic Functions. CRC Press, Inc.

 Sivaram K. Narayan, Daniel Sievewright, and Derek Thompson (2016). Complex Symmetric Composition Operators on H2. Journal of Mathematical Analysis and Applications, 443(1): 625-630.

 James E. Tener. (2010). Unitary Equivalence to a Complex Symmertric Matrix. Pomona College, Claremont, CA.